Two-Gate Extensions of Free Axis and Free Quaternion Selection for Sequential Optimization of Parameterized Quantum Circuits

Abstract

We propose two-gate extensions of the sequential single-qubit optimizers, Free Axis Selection (Fraxis) and Free Quaternion Selection (FQS), termed Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), respectively. In contrast to Fraxis and FQS, which update one single-qubit gate at a time via quadratic local cost function and matrix diagonalization, TGF and TGFQS optimize two parameterized single-qubit gates simultaneously by constructing an exact quartic local cost function and optimizing it using classical optimizers. We further investigate how different gate pairing strategies affect optimization performance. Using numerical experiments on spin Hamiltonians, molecular Hamiltonians, and quantum state preparation tasks, we find that TGF and TGFQS frequently achieve a lower final relative error to the ground state energy or infidelity than their single gate counterparts. We observe that the random and half-shifted gate pairing strategies for TGF and TGFQS perform best in many of the tested settings. In the additional finite-shot experiments on Fermi-Hubbard and transverse-field Ising model Hamiltonians, the best gate pairing strategies retain their advantage across the tested shot counts in shallow circuits. These improvements come at the cost of increased circuit evaluations per gate update, highlighting a trade-off between the power of local optimization and measurement overhead.

Figures (11)

• Figure 1: Single-qubit gate tomography for sequential optimization with Fraxis and FQS.
• Figure 2: Tomography for two parameterized single-qubit gates for two-gate optimization with TGF and TGFQS.
• Figure 3: Hardware-efficient ansatz consisting of parameterized single-qubit gates $R_i$ for $i =1,2,\ldots, D$ followed by an entangling layer consisting of controlled-Z gates for every qubit pair, repeated over $L$ layers for $n$ qubits.
• Figure 4: Results for 4-qubit Fermi-Hubbard model on a $1\times2$ lattice for standard Fraxis and FQS (red) as well as TGF and TGFQS with the following gate pairs: linear (blue), random (green), opposite (black), and half-shifted (orange). In both figures, the number of layers was set to $L=4$, each line represents a mean of 20 runs, and the shaded areas are 68% confidence intervals around the mean. The figures are shown on a semi-log scale, with the vertical axis showing the relative error with respect to the ground state energy.
• Figure 5: Results for 4-qubit Fermi-Hubbard model on a $1\times2$ lattice for standard Fraxis and FQS (red) as well as TGF and TGFQS with random gate pairing (green). In both figures, the number of layers was set to $L=2$, and each line represents a mean of 20 runs. The line styles indicate the number of shots: solid for 4096 shots, dashed for 8192 shots, and dash-dotted for 16384 shots. The vertical axis shows the relative error with respect to the ground state energy.